No, not in general. Take $C=\mathbb{P}^1$, $L=O(1)$, $p$ to be map $x\mapsto x^2$ in affine coordinates. Then $p_*L$ has rank $2$, but $h^0(p_*L)=2$, so $e_1$ and $e_2$ can't both be positive.
Donu Arapura
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